Numerical simulations of UV wind-line variability in magnetic B stars:βcephei

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1 Astronomy & Astrophysics manuscript no. zeus c ESO 7 December, 7 Numerical simulations of UV wind-line variability in magnetic B stars:βcephei R. S. Schnerr,, S. P. Owocki 3, A. ud-doula 3, H. F. Henrichs, and R. H. D. Townsend 3 SRON, Netherlands Institute for Space Research, Sorbonnelaan, 38 CA Utrecht, the Netherlands roalds@sron.nl Astronomical Institute Anton Pannekoek, University of Amsterdam, Kruislaan 3, 98 SJ Amsterdam, Netherlands 3 Bartol Research Institute, University of Delaware, 7 Sharp Lab, Newark, DE 976, USA Received date/accepted date ABSTRACT Context. Winds of many early-type stars studied in the UV by the satellite show cyclic or even periodic variability on a rotational timescale. In recent years a number of such stars were found to have magnetic fields with polar field strengths in the range of - 3 Gauss. Aims. In an attempt to explain the strictly periodic variability in these stars with relatively weak magnetic fields, we have performed line profile calculations of simple stellar wind models and D-MHD simulations. Methods. For our simulations we have taken the BIV starβcephei as our prototype, as this star has been studied extensively, has a slow rotation rate and has a favourable inclination of the rotation and magnetic field axes. Results. We find that simple models with an enhanced density of absorbing ions in the magnetic equator, qualitatively reproduce the observed variability of the UV wind lines. Such enhancement of density might naturally be expected due to channelling of material towards the magnetic equator by the magnetic field. However, quite surprisingly, we find that less stellar flux is absorbed in the magnetic equator compared to the magnetic poles in the MHD simulations. This is because material in the magnetic equator tends to be either confined to a geometrically thin disk, or have a temperature too high for the ions of typical UV wind lines, resulting from shock heating due to the colliding winds of the two magnetic hemispheres. Conclusions. Although magnetic channelling of the stellar wind is certainly important, we conclude that the inclusion of X-ray ionisation is likely required to explain the observed UV wind line variability. Key words. MHD, Line: profiles, Stars: magnetic fields, Stars: winds, Stars: mass-loss, Ultraviolet: stars. Introduction The majority of the O stars and a significant fraction of the early B stars exhibit variability in their UV wind lines. In well-studied cases this variability has been found to be either strictly periodic or cyclic, i.e. not phase locked from one year to the next (see for instance Fullerton 3, for a review). In the chemically peculiar Ap/Bp stars, which have kg magnetic fields, the strictly periodic variability can be explained by the stellar rotation of the magnetic field which modulates the outflow. Pointed out by their strictly periodic UV wind line variability, recently several non-chemically peculiar OB-type stars were also found to posses magnetic fields. These are the B starsβcep (Henrichs et al. ),ζ Cas (Neiner et al. 3a), V Oph (Neiner et al. 3b),τ Sco (Donati et al. 6b) andξ CMa (Hubrig et al. 6), and the O starθ Ori C (Donati et al. ). The B starωori (Neiner et al. 3c) and the O star HD 96 (Donati et al. 6a) have also been found to posses a magnetic field, but ω Ori was only observed to show cyclic variability over a period of three days, and HD 96 was pointed out as a magnetic candidate by strong periodic changes in its optical spectrum (Walborn et al. ). These stars have weaker large scale fields ( 3 G) and stronger winds than the Ap/Bp stars, which are mostly of a later spectral type. The observed strictly periodic variability is most likely related to the presence of these magnetic fields. To characterise the capability of a magnetic field to influence the flow of the wind ud-doula & Owocki () defined the wind magnetic confinement parameter η = B eq R /Ṁv, where B eq is the magnetic field strength at the magnetic equa- 3 tor, R the stellar radius, Ṁ the mass loss rate, and v the terminal velocity of the wind. For the strongly magnetic Ap/Bp starsη is of order 3 or more and the magnetic field completely dominates the wind flow up to several stellar radii from the star. In the case of the OB stars with weaker magnetic fields 3 and stronger stellar winds,η is of the order of and magnetic fields will still play an important role but no longer completely determine the flow. For early-type stars without a detected magnetic field and for which the variability has been found to be cyclic, i.e. not strictly periodic, the origin of the variability is unex-

2 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei plained. Henrichs et al. () found that three different kinds of variability can be distinguished: the Discrete Absorption Component (DAC) type (absorption at high blue shifted velocities), the magnetic oblique rotator type (absorption around zero velocity) and an intermediate type (absorption at intermediate velocities), and concludes that they are all likely to be related to magnetic fields. The timescales for the DAC- and oblique rotator type are of the order of a few days to weeks, i.e. the rotational timescale, whereas the timescale of the intermediate type remains undetermined because of a lack of time coverage. Stellar pulsations can also cause variability in UV wind lines, but the observed timescales are shorter than the rotational timescale. Using phenomenological models and D-MHD simulations, we have investigated whether the presence of the measured 3 G magnetic fields can explain the strictly periodic variability observed in the UV wind lines of B stars. Understanding the origin of such strictly periodic variability could help us to better understand the variability observed in other stars that has not been found to be strictly periodic. Most of the UV wind-line variability of B stars is observed in the lines of SiIV, CIV and NV. At the relevant effective temperatures of up to 3. K, the dominant ionisation stage is always well below these observed lines (Arnaud & Rothenflug 98). The fact that we do observe lines of these ions is thought to be due to superionisation by X-rays. The precise origin of the X-rays is unknown, but excess X-rays are observed for many OB stars (e.g. Berghöfer et al. 996). As a test case we have modelled the bright BIV starβcep, of which the parameters are known relatively well and which has a favourable geometry with both magnetic poles (almost) crossing the line of sight each rotation period.. The UV behaviour and magnetic field ofβcep The stellar parameters of the BIV starβcep (HD, V=3.) are summarised in Table. A clear 6 or day period in its UV lines was reported by Fischel & Sparks (97). Henrichs et al. (993) proposed that this day period was the rotational period, which was seen in the wind lines due to a co-rotating magnetic field that affected the wind. This was confirmed by the discovery of the magnetic field by Henrichs et al. (, see also Donati et al. ). The equivalent width (EW) of the C IV doublet at 8.3/.777 Å and the magnetic field measurements folded with the rotation period of.7 days are shown in Fig.. The magnetic extrema (minimum and maximum field strength) coincide with phases of minimum CIV EW and phases of zero field strength coincide with maximum CIV EW. This behaviour is indeed expected from a dipole magnetic field not strong enough to completely dominate the flow, but which does influence the wind up to a few stellar radii. As a result the stellar wind is guided towards the magnetic equator until it becomes to weak to further confine the wind. One would expect such a scenario to result in enhanced absorption in the magnetic equator and reduced absorption over the magnetic poles. From modeling of the rotational variability of the magnetic field strength the angle between the rotation axis and the magnetic axis (β) could be determined (Donati et al. ). Although both i andβare large, 9, they are not very strongly constrained. The inequality of the two C IV EW min- ima, and the slight offset of the average magnetic field strength visible in Fig. imply for a dipole field that the inclination cannot be exactly 9. However, to keep the interpretation of the simulations as simple as possible, we have assumed both i and β to be 9, which is an equator-on star with the magnetic axis in the rotational equator. In the spectra of this star regular Hα outbursts have been observed since 933 (Karpov 933, see Pan ko & Tarasov 997 for an overview of emission phases until 99). However, the Hα emission showed no evidence of any rotational modulation, which is very difficult to understand in the presence of a dipole magnetic field with its axis in (or close to) the rotational equator. If the emission would originate from a magnetically confined disk, comparable to what is seen in some strongly magnetic Ap/Bp stars (see Townsend et al. ) strong rotational modulation should definitely be observed. Motivated by these considerations, the system was closely examined by Schnerr et al. (6) with spectroastrometric techniques, which has lead to the discovery that the emission originates not from the primary star but from the close companion which is in a 9 year orbit and which has been regularly observed by speckle interferometry. B long (G) EW(C IV) [ 7, 8]km/s P=.7() days T min =976.(6) , 8 spectra TBL 998, 8 spectra UV phase Fig.. The EW of the CIV doublet ofβcep as a function of rotational phase for a rotation period of.7 days (top) and the magnetic field measurement from 998- folded with the same period (bottom). Figure taken from Henrichs et al. (). 3. Line profile calculations using SEI In radiation transfer of line driven winds, the Sobolev approximation is often used. The great advantage of this method is that it allows for an estimation of important parameters, such as optical depth, based on the local physical conditions only. For the calculation of the line profiles the Sobolev approximation is used in two different contexts. First to calculate the

3 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei Norm. Flux ( erg cm s σ Å obs /σ ) exp.8 Wavelength (Å) 3 spectra Norm. Flux σ obs /σ exp Wavelength (Å) 3 spectra Norm. Flux ( erg cm s Å ) σ obs /σ exp 3..8 Wavelength (Å) 3 spectra Quotient flux σ obs /σ exp Velocity (km s ) (stellar rest frame) Wavelength (Å) 3 spectra Quotient flux σ obs /σ exp 6 Velocity (km s ) (stellar rest frame) Wavelength (Å) 3 spectra Quotient flux σ obs /σ exp Velocity (km s ) (stellar rest frame) Wavelength (Å) 33 spectra Velocity (km s ) (stellar rest frame).6.. Velocity (km s ) (stellar rest frame).6.. Velocity (km s ) (stellar rest frame) Fig.. UV wind line profiles of β Cep normalised by the average line profile, as observed with the satellite of Si IV (left), C IV (middle) and NV (right). A clear modulation is visible with the rotation period of. days. Table. Stellar properties ofβcep, taken from Henrichs et al. () and Donati et al. (). M M R 6. R log(l /L ). P rot. days T eff 6 K Spectral type B IV B polar 36 G i 9 β 9 impact parameter p star r z observer 3 3 source function throughout the wind, and second to reduce the integrals along sight-lines required to solve the transfer equation for a given frequency, to a local problem in the resonance zone. It was pointed out by Hamann (98) that most of the deviations in this approximation are due to solving the transfer equation and not to the calculation of the source function. Fig. 3. The integrations for calculating the line profiles are performed along sight lines (z) with a fixed impact parameter p. This motivated Lamers et al. (987) to develop a method that exploits the efficiency of the Sobolev approximation, but is more accurate. This method, Sobolev with Exact Integration

4 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei (SEI), uses the Sobolev approximation to calculate the source function, but solves the transfer equation by direct integration. For all our line profile calculations we have used the SEI routine developed and described by Cranmer & Owocki (996). This routine is designed to calculate line profiles for arbitrary geometries in, and 3 dimensions. We have assumed atomic parameters for the 8 Å C IV line and neglected interaction between the doublet members and limb-darkening. 3.. Solving the transfer equation in the comoving frame Using the escape probability method introduced by Castor (97), we calculate the source function S ν in the Sobolev approximation as (assuming that the ratio of collisional over radiative de-excitations ǫ = ): S ν = I c β c β, () where I c is the flux of the star,β = P esc (µ,φ) the angleaveraged escape probability andβ c = D(µ)P esc (µ,φ) is the core penetration probability, where D(µ) = for rays intersecting the star and D(µ)= otherwise. The variablesµ (= cosθ) andφdenote a direction from a point in the wind, relative to the local radial direction. The photon escape probability is calculated as P esc (µ,φ)= e τ τ, withτ=χv th /(d[v a]/da), where d[v a]/da is the derivative of the velocity component along the direction a, which is set byµandφ. To calculate the integrals required to determine the angle averaged escape probability and all other integrals required for the line profile calculations, we have used Romberg s method of numerical integration described in Press et al. (99) with an estimated fractional accuracy of.%. The line profiles are calculated by numerically integrating the formal solution to the transfer equation along sight-lines with a fixed impact parameter, as illustrated in Fig. 3, for many different sight-lines. The number of sightlines used is increased until the required precision of.% of the flux is achieved. These integrals have to be performed for all relevant wavelengths or velocities; in our case equidistant velocity points in the to+3 km s range. The local optical depth dτ=χdz is determined by χ(r, v,ν)= πe g g f n Civ Φ(n z v,ν), () m e c ν D g l with πe m e c =.6 cm s, ν D =ν v th /c withν the line frequency, v th the ion thermal speed, c the velocity of light, g g f =.38, g l =, n Civ the CIV number density, and n z a unit vector along the sightline. The local line profileφ(n z v,ν) is assumed to be given by a normalised Gaussian Φ(n z v,ν)= e (n z v/v th ) π, (3) where n z v/v th is the projected velocity offset relative to the centre of the line in units of the thermal velocity.. A phenomenological model From the observed variability in the UV line profiles shown in Figs. and, it is clear that the absorption is reduced when we see the magnetic poles, and enhanced when we see the magnetic equator. The most straightforward way to explain this 9 would be an enhanced density in the magnetic equator relative to the magnetic poles. Such an enhancement of the density in the equatorial regions could be expected due to the channelling of the wind by a dipole-like magnetic field which tends to guide the wind towards the magnetic equator. 9 Before exploring more detailed models, it is useful to examine to what extent a very basic model with enhanced density in the equatorial regions would qualitatively reproduce the observed behaviour. For this purpose we have calculated line profiles for a model that has an artificially increased density near the magnetic equator. As we have not included rotation in our models, we have defined our polar coordinates relative to the magnetic axis for all models discussed in this paper. In this geometryθ is defined as the longitude relative to the magnetic axis, i.e.θ= and θ=8 define the magnetic poles, andθ=9 denote the magnetic equator. Rotation is simulated by adjusting the viewing angle of the observer, where the phase is for an observer at [θ=, r ]. For our model the mass loss rate of the star scales with the sin θ, whereθ is the azimuth angle relative to the magnetic axis. We have assumed a CAK wind (Castor et al. 97), with aβ-law velocity as a function of radial distance r: v(r)=v ( R /r) β, () withβ=.8, v = km s and R is the stellar radius. The density in the wind can then be written as ρ(θ, r)= Ṁ(θ) πr v(r), () where we have parametrised the mass-loss rate as Ṁ(θ)= Ṁ sin θ, (6) with Ṁ =.7 8 M /yr. The dependence of the mass-loss rate onθ is shown in Fig.. At the effective temperatures of B stars of up to 3, K, all carbon is expected to be in the form of CII (Arnaud & Rothenflug 98). The fact that CIV lines are observed is thought to be due to superionisation by X-rays. However, the precise fraction of C IV that is produced by X-rays is unknown, as is the exact origin of the X-rays. For this simple model we have assumed that the X-ray ionisation is constant throughout the wind, with a fixed ionisation fraction of CIV/C=. and a solar carbon abundance of n C /n=.7 (Lodders 3). 3 The results of our line-profile calculations are shown in Fig.. One has to keep in mind that our calculations are for a singlet only. Although there are differences in the detailed line shapes, the main phase dependence of both the emission and absorption is reproduced. Maximum absorption is observed at 3 the magnetic equator, and minimum absorption over the magnetic poles. As a result the whole profile appears to shift up and down, resembling the behaviour of the observations.

5 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei Table. Summary of the basic parameters of the performed simulations. Shown are Q, the mass-loss rate of the non-magnetic models used to initialise the simulations, the terminal velocity of the nonmagnetic models, andη for a dipole magnetic field with a polar strength of 36 G. model Q Ṁ D v,d η 9 M /y km s low-ṁ intermediate-ṁ high-ṁ Fig.. Mass-loss rate as a function of azimuth angle (θ) for the phenomenological model where the mass loss scales with sin θ. Similar behaviour might be expected in the presence of a magnetic field, due to magnetic channelling of the wind. Therefore these results encouraged us to carry out full D- MHD simulations, from which we would expect qualitatively similar results. Norm. flux σ obs /σ exp Wavelength (Å) 3 spectra Velocity (km s ) (stellar rest frame) Quotient flux σobs /σ exp Wavelength (Å) 3 spectra Velocity (km s ) (stellar rest frame) Fig.. Normalised line profiles (left) and line profiles divided by the average line profile (right) of the sin -model. Shown are a measure of the variability (top), the line profiles (middle) and a greyscale plot of the line profiles vs. rotational phase. The main phase dependence of both the emission and absorption observed inβcep is reproduced. velocity equals the Kepler velocity is r R. Instead, rotation of the system was accounted for by calculating line profiles 6 for different viewing angles. The radiative driving is incorporated in the form of the Qformalism as described by Gayley (99). We have performed a series of simulations assuming a different efficiency of the radiative driving mechanism, determined by Q, and hence differ- 6 ent mass-loss rates. The characteristic parameters of three representative models are shown in Table. The CAK parameters α andδused for the radiative driving were set to. and., respectively. The MHD-simulations were initialised from the relaxed solution of a simulation with the same parameters, but 7 without magnetic fields. These simulations were also run for seconds, although a stable solution was always reached within several flow times. From the results of our simulations we calculate the line profiles using the SEI method... Evolution of the simulations 7 Snapshots of some typical models and geometries are presented in Figs. 6 and 7. The main characteristic of all runs is that they are highly variable. Quasi-stationary behaviour is observed once the influence of the initial conditions has disappeared, but no stable situation is reached. Guided by the mag- 8 netic field, material builds up in regions around the equator, and then either breaks out or falls back onto the star. A sketch of the important regions is shown in Fig. 8.. D-MHD simulations Our implementation of the generic ZEUS based code has been described in detail by ud-doula & Owocki (, see also ud- Doula 3), with modifications to include an energy balance with radiative cooling (ud-doula 3; Gagné et al. ). Using this code we compute the relevant physical parameters in a non-equidistant mesh of zones coveringθ from to 8 by 3 zones covering r from to R. The total simulated time for a typical simulation is seconds ( 6 days), and all relevant system parameters are saved every seconds (.8h), giving snapshots per simulation. With the adopted stellar parameters from Table the characteristic flow time of the system is of the order of R max /v R /=3 s. Since β Cep is a very slow rotator the dynamical effects of rotation could be neglected. The radius at which the co-rotation star "hot loop" collision region outflowing disk Fig. 8. Sketch of the most notable regions observed in the simulations. The hot loop. A hot loop of low density plasma is clearly visible in the temperature plots, but also in the density plots. 8 The size of the loop depends strongly on the mass-loss rate. In

6 6 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei Fig. 6. Snap-shots of the simulations. Shown are the evolution of a typical simulation (d) of the density (log ρ [g/cm3], top) and temperature (log T [K], bottom) after, 3,, and s the low-ṁ model it extends to almost a stellar radius above the stellar surface. In the intermediate-ṁ model the loop is only a fraction of a stellar radius in height and in the high-ṁ model it has completely disappeared. This is most likely related to the higher density in the intermediate- and high-ṁ models, which increases the efficiency of the radiative cooling. The collision region. Near and below the tops of the last closed field lines in the equator material is captured by the magnetic field lines and heated by shocks created by the inflowing wind. Density builds up here until it either falls back towards the star or breaks the magnetic field lines open and is blown away by the radiative driving. Some of the material that falls back towards the star is guided towards higher latitudes by magnetic field lines closer to the star and blown away at midlatitudes by the stellar radiation. The location and size of this region is most clearly seen in the density and temperature plots. The extent of the region is determined by the maximum radius at which the magnetic field is still able to determine the wind flow. As a result this region is smaller for the models with a higher mass-loss rate. The outflowing disk. Connected to the collision region is a hot, high density outflowing disk, which has a low outflow velocity due to the high densities. The material in this disk comes from material that breaks open the magnetic loops in the col- lision region and wind of higher latitudes that is still able to guide the material towards the equator although it is too weak to dominate the flow. The scale height of the disk is set by the ability of the shocked material in the disk to cool and contract. As the efficiency of the radiative cooling depends strongly on the density, the disks are thinner for a higher mass loss and more puffed-up when the mass loss is lower. Layers of the disk that are able to cool efficiently form thin, variable, outflowing sheets. The dynamics observed in these full D-MHD simulations 3 is quite different from the simple enhancement of the density towards the magnetic equator as discussed in Sect.. Material from both hemispheres is guided towards the equator, but due to the shock-heating of the gas the temperatures are much too high to allow a large fraction of CIV to be present in an ex- 3 tended region near the magnetic equator. Material that is cool enough to have a significant fraction of CIV in the equatorial region, is confined to thin sheets that at a given time only cover a small fraction of the stellar disk. Therefore it is not clear that the absorption along the equator is actually increased compared 33 to that over the poles... Calculating line profiles from D-MHD models The CIV observed in the stellar winds of B stars is thought to be related to superionisation by X-rays. As the origin of the X- rays is not completely understood, the precise fraction of C IV 33 that is produced is unknown. To determine the CIV density as a function of temperature we have used a gaussian fit to the calculated C IV fractions by Arnaud & Rothenflug (98, see Fig. 9). To mimic the production of C IV by X-rays throughout the wind we have assumed a floor value of 3% at the low tem- 3 perature end. We have adopted the solar abundance of carbon of n C /n=.7 from Lodders (3).

7 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei 7 Fig. 7. Snap-shots of the simulations. Shown are (from left to right) density (logρ [g cm 3 ]), temperature (log T [K]), radial velocity (km s ) and transverse velocity (km s ) after s for the low, intermediate, and high-ṁ models (top to bottom). 3 Fig. 9. A gaussian fit to the calculated ionisation fractions of CIV by Arnaud & Rothenflug (98), including a minimum ionisation fraction of 3% at the low temperature end to mimic the production of CIV by X-ray ionisation. To calculate line profiles from our simulation we have timeaveraged the C IV density to remove all temporal variability and account for the fact that our simulations have no structure in the in the plane of stellar rotation. We have averaged the CIV density for each gridpoint for the last 3 snapshots, when all effects of the initialisation have faded and quasi-steady behaviour is observed. The radial and transverse components of the velocity at each gridpoint are determined by a weighted average of 3 the velocity components with the CIV density. The highest CIV density near the magnetic equator is found in the cool, variable, outflowing disk, which only covers a fraction of the stellar disk. As a result of the time averaging, the relatively high density of CIV ions in this outflowing disk will be spread out over a larger 3 area near the magnetic equator. This will tend to increase the total absorption over the equator, which, although perhaps not the most accurate approach, gives us the best chance of reproducing the observed behaviour. From these average parameters we have calculated line pro- 36 files using the SEI method. Examples of the line profiles of the low, intermediate and high-ṁ model are shown in Fig.. The EW of the line and of the emission and absorption parts are shown in Fig..

8 8 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei Quotient flux Norm. flux σ obs /σ exp Wavelength (Å) 3 spectra Velocity (km s ) Quotient flux Norm. flux σ obs /σ exp Wavelength (Å) 3 spectra Velocity (km s ) Quotient flux Norm. flux σ obs /σ exp Wavelength (Å) 3 spectra Velocity (km s ) Fig.. Results of the line profile calculations of the low (left), intermediate (middle) and high-ṁ (right) models. Line profile were calculated from the averaged CIV density of the last 3 snapshots, assuming a low temperature floor of 3% for the fraction CIV/C Contrary to what we expected, the behaviour of the C IV is quite different from what is observed and what was reproduced with our simple phenomenological model. Instead of increased absorption, the absorption has actually decreased in the equatorial regions. This is due to the high temperatures in the equatorial regions, and the small height of the outflowing disk of material that has been able to cool. In the observations both the red and blue parts of the line show a higher (or lower) flux simultaneously. The line profiles of all the simulations show an anticorrelation between the emission and absorption part of the line profile: enhanced absorption in the blue wing of the line coincides with enhanced emission in the red wing and vice versa. When we compare the higher and lower flux phases of the red and the blue wings of the simulations with the observations, we see that the red part of the line shows similar variability with phase: higher flux near the magnetic poles and lower flux near the magnetic equator. The blue part of the line shows the opposite behaviour..3. Understanding the line profiles Due to the magnetic field, mostly material from midlatitudes is channelled towards the magnetic equator. As the magnetic field lines are almost radial over the poles, magnetic channelling is less effective in these regions. This effect is amplified by the radiative driving that is able to accelerate the less dense material at the midlatitudes to higher velocities. As a result for a given radius the wind density first decreases with increasing θ when going from the magnetic pole towards the midlatitudes, and then increases towards the magnetic equator. However, due to the high temperatures in the equatorial region, the C IV density can still be quite low. In the magnetic equator the density is higher and radiative driving is less efficient, resulting in lower radial velocities. In Fig. 7 we can see that the radial velocities in the outflowing disk are of the order of 3 7 km s, where the low mass-loss models have lower speeds than the high mass-loss models. So although the CIV density may not be very high in the equatorial regions due to the temperatures, almost all of the material is absorbing photons at low velocities and no photons are absorbed at high velocities. This is reflected in the spectra as a function of phase. At the poles the wind is not much affected by the magnetic field. Going to the mid-latitudes densities decrease somewhat, as material has been guided towards the equator. As a result the velocities are higher and the absorption decreases compared to the poles. Closer to the equator the collision region and the region between the hot loop and the collision region begin to cover the star. As the radial velocities are very low there and some material is even falling back onto the star, photons are only absorbed at low velocities and not at higher velocities. This results in more absorption at low velocities and less absorption at higher velocities near the equator as compared to the poles. 6. An X-ray ring model The observations ofβcep show enhanced CIV absorption over the magnetic equator and reduced absorption over the magnetic poles. This behaviour is reproduced by our phenomenological model with enhanced density in the equatorial region, dis- cussed in Section. We expected that magnetic channelling of the wind would naturally reproduce similar behaviour, but the D-MHD models described in Sect. instead show reduced absorption near the equator relative to the poles. Since magnetic channelling does not seem to be able to explain the observed behaviour we have to consider alternative models. Up to now we have assumed a fixed ionisation fraction in the wind. Depending on the process producing the ionising X-

9 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei 9 Fig.. Equivalent width of the absorption (middle), emission (right) and total lineprofile (left) in Å assuming the wavelength of the 8 CIV line as a function of rotational phase. 3 3 rays, the ionisation fraction could vary significantly throughout the wind. One process that is likely to produce X-rays, is the stellar wind from one (magnetic) hemisphere that is channelled towards the magnetic equator by the magnetic field and collides with the wind of the other hemisphere. In the shock region that results from this collision, the wind can reach temperatures of up to 8 K. The strongest shock region will be at the largest radius where the magnetic field is still able to channel the stellar wind. At this radius the winds of the two hemispheres will collide with the highest velocity. This radius is typically the Alvén radius, which forβcep (withη ) is of the order R. To simulate the line profiles that would be observed from a star where CIV is formed by ionisation of X-ray originating from a thin ring, we have used the toy model described in App. A. In this model the X-ray emission from the ring is assumed to be optically thin, and the ionisation fraction is assumed to be proportional to the X-ray intensity over the electron density. The spherically symmetric wind structure is defined by a β-law velocity law withβ=.8 and v = km s and a mass-loss rate of.3 8 M year. X ray ring Fig.. The geometry of our X-ray ring model. The radius of the ring is set by b. We have calculated line profiles from our X-ray ring model using our SEI code. Reasonable variability of the line profiles is only found for simulations where the radius of the ring is significantly smaller than the Alvén radius. Example line profiles of a simulation with b=.3 R are shown in Fig. 3. This radius of the ring is similar to that of the hot loops observed in the MHD simulations. z b y x Although this is a simplified model, the results are very encouraging. The line profiles have many similarities with the observed line profiles ofβcep in Fig.. The main phase dependence of both the absorption and the emission is reproduced, 6 and the same double-peaked structure is visible in the amplitude of the variability shown in the top panel. A configuration as described here, where the X-rays originate from a non-rotationally symmetric geometry (as the axis of the magnetic field is in general different from that of the rota- 6 tional axis) would result in rotationally modulated X-ray flux. The modulation expected for our X-ray ring model is calculated in App. B. Modulated X-ray flux is indeed observed for the O7.IIIe starξ Per (Massa et al. ). Normalized flux σobs /σ exp Wavelength (Å) 3 spectra Velocity (km s ) (stellar rest frame) Quotient flux σobs /σ exp Wavelength (Å) 3 spectra Velocity (km s ) (stellar rest frame) Fig. 3. Line profiles for a wind where the number of absorbing ions is determined by the balance between the ionising flux from an X-ray ring and recombinations with free electrons. 7. Conclusions and discussion 7 In an attempt to explain the line-profile variability that is observed in the UV wind lines of known magnetic massive stars, we have performed line profile calculations of toy-models and full D-MHD simulations of the stellar wind structure. The observations of the UV wind lines show enhanced ab- 7 sorption over the magnetic equator compared to the magnetic poles. A simple phenomenological model with enhanced density near the magnetic equator qualitatively reproduces this be-

10 R. S. Schnerr et al.: Numerical simulations of UV wind-line variability in magnetic B stars:βcephei haviour. Such an enhancement of density in the equatorial regions could result from the magnetic channelling of the wind by the magnetic field. However, when we try to reproduce this behaviour with full D-MHD simulations, these results are not confirmed. Due to shock heating of the gas, most material near the equator is too hot to contain much CIV. The material in the equatorial regions that is able to cool is confined to thin sheets, that cover only a fraction of the stellar disk. As a result, we find that in fact the absorption near the equator is reduced instead of increased, contrary to what is observed. It seems that a detailed treatment of the superionisation due to X-rays in the stellar wind is required to explain the observed behaviour. Our simple model of a thin X-ray ring in the magnetic equator is already able to qualitatively reproduce the main characteristics of the observed variability. This could also explain why the terminal velocity observed in the wind lines of 7 km s is lower than the theoretically predicted terminal speed. Perhaps the X-rays are produced relatively close to the star, and able to maintain a higher fraction of superionisation close to the star than further out in the wind. An alternative explanation could be that in B-type stars the outer parts of the wind remain invisible due to the lower mass loss rates (as compared to O stars). Appendix A: X-ray emission from a ring We estimate the ionisation fraction due to X-rays emitted from an optically thin, infinitely thin ring of radius b in an optically thin wind, as shown in Fig.. For this purpose, we calculate the mean intensity of X-rays at each point in the wind. In standard spherical coordinates a point in the xz-plane, as seen from a point on the ring are given by: r = r b where b=(b cosφ, bsinφ, ) is a point on the ring, and r=(r sinθ,, r cosθ) (A.) (A.) (A.3) is a point in the wind in the xz-plane The distance of this point in the wind to a point on the ring is given by: r = b + r rb cosφ sinθ (A.) To calculate the mean intensity, we integrate the flux over the entire ring: dl F(r,θ)= F r = F b b + r π dφ a cosφ, with F the X-ray emissivity per unit length of the ring and a= rb sinθ b + r. (A.) (A.6) As r> and b> this means that <a. For this interval we can evaluate the integral as: F(r,θ)= F b b + r π a (A.7) For points on the ring a= and we have a singularity, as we have assumed that the flux decreases as /r and the ring is infinitely thin. As this concerns only a very small fraction of the points, we can solve this by setting a maximum value on a of.99. If we assume a simple two-level atom, the ionisation balance is determined by the ratio of X-ray ionisations to recombinations 3 n upper n lower F(r,θ) n e, (A.8) which gives the fraction of atoms in the upper level, as n upper + n lower = n for a two-level atom. Appendix B: X-ray variability due to star occultation 3 In the model presented in Sect. 6 and Appendix A, where the X-rays originate from an infinitely thin ring, it is possible to calculate the expected X-ray lightcurve assuming that no X- rays are absorbed in the stellar wind. Occultation of a part of the X-ray ring by the star will result in variability. From the point of view of the observer, the star casts a shadow which has the shape of a cylinder with a radius of R to the backside of the star (see Fig. B.a). The observer is assumed to be in the direction of x and the star at the origin. The X-ray ring will disappear in the shadow behind the star, somewhere on the intersection of the surface of this shadow-cylinder and a sphere with radius b, the radius of the ring. This intersection has the shape of a circle with a radius of R around the x-axis at x=a= b R (see Fig. B.b). For a given rotation phase of the star, the plane of the ring can be chosen to be parallel to the z-axis. The angleφ(see Fig. B.b) is then determined by the rotation axis and rotation phase of the star. Only ifα<φ<π α (φ [,π]) part of the ring will be in the shadow of the star. From Fig. B.c we can see that p=a/ sinφ, which gives us the angleβ=arccos p/b. The total fraction of the ring that is occulted is β/π = arccos(p/b)/π. If the X-ray ring is located in the magnetic equator, the angleφcan be calculated using e x B/ B =cosφ. Fig. B. shows lightcurves for different b, with the rotation 6 axis perpendicular to the line of sight, and a magnetic field axis which has an angle of 9 degrees with the rotation axis. Acknowledgements. RSS thanks A. de Koter for useful discussions and S. Cranmer for making his SEI code available. References 6 Arnaud, M. & Rothenflug, R. 98, A&AS, 6, Berghöfer, T. W., Schmitt, J. H. M. M., & Cassinelli, J. P. 996, A&AS, 8, 8

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